Abstract.
The study of the spectrum of the Laplace operator on a
hyperbolic surface is an important topic which has
numerous applications
for example in number theory and
geometric equidistribution problems.
The discrete eigenfunctions in this setting are called
\textit{Maass waveforms}
after Hans Maass who first studied them (1949).
Of special interest are so-called \textit{arithmetic}
hyperbolic surfaces; their spectra have
several peculiar properties, one of which is
the often occurring Maass waveforms of
eigenvalue exactly $\frac 14$, related to
certain \textit{Galois representations}.
In some cases the existence of these Maass waveforms
is still only conjectural
(it would follow from a conjecture of Artin),
but several important cases have been settled,
in particular by results of Langlands and Tunnell (1980-81).
The theorems of Langlands and Tunnell also play a small but crucial
role in Wiles' proof (1995) of Fermat's Last Theorem.
In my talk I will attempt to explain some aspects of
these matters in a way
accessible to a general mathematical audience.
I will also briefly describe some recent spectral computations by
Andrew Booker (University of Michigan) and myself, where we
were forced to construct several new
explicit examples of Galois representations related to Maass
waveforms.